The main cases of using the preposition at

At
At is one of the oldest prepositions: it has been found in literature since 755 as an indication of a certain “point”, i.e. minimum space, place of making and completing the movement. The Old English preposition at is related to the Latin ad k and, in its basic spatial meanings, corresponds to the Russian prepositions k and u. Since 1000, the use of at has been noted to indicate employment in a particular place
The main cases of using the preposition at:
To designate a place:
a) with the value of y, near, about:
She is standing at the window. Она стоит у окна.
Не stopped at the door. Он остановился около двери.
b) with the value in before the names of small towns, villages, etc. (to the question where?):
Не was born at Klin. Он родился в Клину.
c) with the meaning in, on before the names of institutions, organizations, meetings of people, i.e. when indicating the place where any action is performed, any process (to the question where?)
I will see him at the theatre tonight. Я его увижу сегодня вечером в театре (т. е. на спектакле).
The meeting took place in the theatre. Собрание состоялось в театре (т. е. в помещении театра.
Не works at a factory (at an office, at an institute). Он работает на фабрике (в конторе, в институте).
I will see him at the lecture (at the lesson, at the meeting). Я увижу его на лекции (на уроке, на собрании).
In
In is one of the Old English prepositions (it has been found in literature since 700) and plays a very important role in the modern language. It is used to convey a wide variety of spatial, temporal and abstract relationships. In Old English, in indicated not only an action or stay in a certain place, but also a movement penetrating into a closed space (now into).
In occupies a place in English corresponding to the place of the preposition in in Russian.
To indicate a place with a value in (to the question where?):
The pencil is In the box. Карандаш в коробке.
We live in Moscow. Мы живем в Москве.
Into
The preposition arose in the 10th century from the combination of in and to and is a formation of the same type with up to and out of and other combinations, of which the first component is an adverb and the other is a preposition. Prior to this, to indicate inside a closed space, in was used with the accusative case. In was originally understood as an indicator of the general direction of movement, and to as a qualifier, and in this case, since both words expressed one concept, they began to be pronounced (and written) together as one word , with the stress shifted to the first syllable.
The primary meaning into refers to incoming traffic crossing the outer boundary of a particular enclosed space or space. Into is used without exception with all verbs of penetrating motion.
The preposition Into is used with the meaning in and denotes the direction of action inside something (to the question where?):
I am going into the room. Я иду в комнату.
Не put the book into his bag. Он положил книгу в свой портфель.

Off
This word comes from the accented adverb of. from the 15th century It became a separate word (and began to be written off), but partially retained the original meaning of: separation, distance (from, from, away from, away from h-l in combination with the concept of a certain surface. The meaning of separation - distance has a preposition in messages as about the spatial relations of concrete objects, and about the relations of abstract concepts

The preposition off denotes the movement of an object from the surface of another object and is usually translated into Russian by the preposition s (so).
Please take your book of the table. Уберите вашу книгу со стола, пожалуйста.
The main cases of using the preposition off:
With the value c when designating an object from the surface of which it is removed, someone is separated. or something. (opposite in meaning to the preposition on):
Не took all the things off the table. Он убрал все вещи со стола.

The rain ran off the roof. Дождь стекал с крыши.

The knife fell off the table. Нож упал со стола.

On (Upon)
On, attested in literature from the 9th century to the present, has retained its main meaning: finding an object on the surface of another object and is usually translated into Russian with the preposition на.
The book is on the table. Книга на столе.
The lexical structure on consists of a large number of meanings associated with the position of a physical or mental position or movement on a certain plane. The physical plane is represented as the surface of the Earth, floor, coast, water mirror, etc. The use of on in this area coincides entirely with the use of the Russian preposition corresponding to it –на.
In relation to place, on means the position or movement of one object on (along) the top, bottom or side surface of another object.

To designate a place with the meaning on in the sense of on the surface (to the question where?) or to the surface (to the question where?):

The picture is hanging on the wall.Картина висит на стене.

Put the magazine on the table. Положите журнал на стол.

out of
The complex preposition out of, which has been found in literature since the 9th century, means from inside when moving from a closed space and outside when staying outside the latter. As part of a complex preposition, the component of has lost its meaning in the composition of the branch, and performs only a connective function, due to which the meaning of the entire combination is conveyed by one out. The antonym of out of in most of its meanings is into.

The preposition out of (from) is used to indicate the direction of action from within, from something (opposite in meaning to the preposition into): Verbs followed by the preposition out of correspond in Russian to verbs with the prefix you:

Не walked out of the house. Он вышел из дому.

He took the letter out of his pocket. . Он вынул письмо из кармана.

He ran out of the room. Он выбежал из комнаты.

Under

In Old English, this preposition, known from written monuments from the ninth century, expressed, as now, various, spatial, temporal and other relationships, generally related to indicating that one object is under another, regardless of the presence of a connection between it.

The main cases of using the preposition under.

Не was lying under a tree. Он лежал под деревом.

Не put the basket under the table. Он поставил корзину под стол.

Under соответствует русскому предлогу под и при выражении некоторых других отношений:

under the guidance (control, man- под руководством (контролем, agement, command) управлением, командой)

under the title (heading) под заглавием

under the name под именем

under fie под огнем (обстрелом)

under the influence под влиянием

Towards

Preposition in its meaning to, towards. found in literature since the 9th century. Both in concrete and in derived abstract meanings, the preposition shows only the direction of the action or feeling towards

The main cases of using the preposition towards

Не was walking towards the sea. Он шел по направлению к морю.

The ship sailed towards the south.Пароход шел по направлению к югу.

This was the first step towards conciliation. Это был первый шаг к примирению.

His attitude towards this matter is favourable.Он благосклонно относится к этому делу.

Не is friendly towards me. Он дружественно настроен ко мне.

The rain stopped towards morning. Дождь прекратился к утру.

We expect to receive the goods towards the end of May. Мы надеемся получить товары к концу мая.